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Y.: Stochastic variational inequalities: Residual minimization smoothing sample average approximations

By Xiaojun Chen, Roger J-b Wets and Yanfang Zhang


Abstract. The stochastic variational inequality (SVI) has been used widely, in engineering and economics, as an effective mathematical model for a number of equilibrium problems involving uncertain data. This paper presents a new expected residual minimization (ERM) formulation for a class of SVI. The objective of the ERM-formulation is Lipschitz continuous and semismooth which helps us guarantee the existence of a solution and convergence of approximation methods. We propose, a globally convergent (a.s.) smoothing sample average approximation (SSAA) method to minimize the residual function; this minimization problem is convex for linear SVI if the expected matrix is positive semi-definite. We show that the ERM problem and its SSAA problems have minimizers in a compact set and any cluster point of minimizers and stationary points of the SSAA problems is a minimizer and a stationary point of the ERM problem (a.s.). Our examples come from applications involving traffic flow problems. We show that the conditions we impose are satisfied and that the solutions, efficiently generated by the SSAA-procedure, have desirable properties. Key words. Stochastic variational inequalities, epi-convergence, lower/upper semicontinuous, semismooth, smoothing sample average approximation, expected residual minimization, stationary point. AMS subject classifications. 90C33, 90C15

Year: 2012
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